Condorcet Method

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					  Condorcet Method

Another group-ranking method
 Development of Condorcet Method

 As we have seen, different methods of
  determining a group ranking often give
  different results.
 For this reason, the marquis de Condorcet,
  a French mathematician, philospher, and
  good friend of Jean-Charles de Borda,
  proposed that any choice that could obtain a
  majority over every choice should win.
          Condorcet Method
 The Condorcet method requires that a
  choice be able to defeat each of the other
  choices in a one-on-one contest.
 Again consider the preference schedules
  from the last section.
           Condorcet Example
 We must compare each choice with every
  other choice:

                      C         D
   A         B

  B          C        B         B

  C          D        D         C

  D          A        A          A
       8         5        6          7
   Condorcet Example (cont’d)
 Begin by comparing A with each of B, C and
 A is ranked higher than B on 8 schedules
  and lower on 18.
 Because A can not obtain a majority against
  B, it is impossible for A to be the Condorcet
                  Choice B
 Next consider B.
 We have already seen that B is ranked higher than
  A, so let’s now compare B with C.
 B s ranked higher on 8+5+7 = 20 schedules and
  lower on 6.
 Now compare B with D. B is ranked higher on 8 +
  5 + 6 = 19 schedules and lower on 7.
 B could be the Condorcet winner since it has a
  majority over the other choices.
                Tabular results
                      The table on the left shows
    A   B   C    D
                       all of the possible one-on-
                       one contests in this
A       L   L    L
B   W       W    W    To see how a choice does
                       in one-on-one contests,
                       red across the row
C   W   L        W
                       associated with that
D   W   L   L
 Although the Condorcet method seems
  ideal, it sometimes fails to produce a winner.
 Try this example!
You Try: Condorcet Example

      A        B        C

      B        C        A

      C        A        B

 20       20       20
  Condorcet Example Explained
 Notice that A is preferred to B on 40 of the
  60 schedules but the A is preferred to C on
  only 20.
 For this reason, there
                 is no
 Condorcet winner.
     Introduction of Paradoxes
 You may expect that if A is preferred to B by
  a majority of voters and if B is preferred to C
  by a majority of voters, then a majority of
  voters would prefer A to C.
 However, we have just seen that this is not
  necessarily the case.
          Transitive Property
 Do you remember the transitive property?
 If we consider the relation “greater than (>)”,
  we see that if a > b and b>c, then a>c.
 However, according to the Condorcet method
  the transitive property does not always hold
 When this happens using the Condorcet
  method, it is known as a Condorcet Paradox.
            Extra Practice
1. The choices in the set of preferences
   shown represent three bills to be
   considered by a legislative body. The
   members will debate two of the bills and
   choose one of them. The chosen bill and
   the third bill will then be debated and
   another vote taken. Suppose you are
   responsible for deciding which two will
   appear on the agenda first.
             Practice (cont’d)

         A            B            C

         B            C            A

                      A            B

    40           30           30

 If you strongly prefer Bill C, which two bills
  would you place first on the agenda? Why?
           Practice (cont’d)

      A             C        B           B
      B             A        C           A

      C             B        A           C
               14        3          22

a. Use a runoff to determine the winner in the
   set of preferences.

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